Year

2012

In Proceedings of the 29th International Conference on Machine Learning, pages: 25-32, ICML ’12, (Editors: John Langford and Joelle Pineau), Omnipress, New York, NY, USA, ICML, July 2012 (inproceedings)

Abstract

Four decades after their invention, quasi- Newton methods are still state of the art in unconstrained numerical optimization. Although not usually interpreted thus, these are learning algorithms that fit a local quadratic approximation to the objective function. We show that many, including the most popular, quasi-Newton methods can be interpreted as approximations of Bayesian linear regression under varying prior assumptions. This new notion elucidates some shortcomings of classical algorithms, and lights the way to a novel nonparametric quasi-Newton method, which is able to make more efficient use of available information at computational cost similar to its predecessors.

Contemporary global optimization algorithms are based on local measures of utility, rather than a probability measure over location and value of the optimum. They thus attempt to collect low function values, not to learn about the optimum. The reason for the absence of probabilistic global optimizers is that the corresponding inference problem is intractable in several ways. This paper develops desiderata for probabilistic optimization algorithms, then presents a concrete algorithm which addresses each of the computational intractabilities with a sequence of approximations and explicitly adresses the decision problem of maximizing information gain from each evaluation.

In pages: 259 -264, IEEE International Conference on Robotics and Automation (ICRA), May 2012 (inproceedings)

Abstract

Performing task-space tracking control on redundant robot manipulators is a difficult problem. When the physical model of the robot is too complex or not available,
standard methods fail and machine learning algorithms can
have advantages. We propose an adaptive learning algorithm
for tracking control of underactuated or non-rigid robots where the physical model of the robot is unavailable. The control method is based on the fact that forward models are relatively straightforward to learn and local inversions can be obtained via local optimization. We use sparse online Gaussian process inference to obtain a flexible probabilistic forward model and second order optimization to find the inverse mapping. Physical experiments indicate that this approach can outperform state-of-the-art tracking control algorithms in this context.

While Gaussian probability densities are omnipresent in applied mathematics, Gaussian cumulative probabilities are hard to calculate in any but the univariate case. We offer here an empirical study of the utility of Expectation Propagation (EP) as an approximate integration method for this problem. For rectangular integration regions, the approximation is highly accurate. We also extend the derivations to the more general case of polyhedral integration regions. However, we find that in this polyhedral case, EP's answer, though often accurate, can be almost arbitrarily wrong. These unexpected results elucidate an interesting and non-obvious feature of EP not yet studied in detail, both for the problem of Gaussian probabilities and for EP more generally.

Latent Dirichlet Allocation models discrete data as a mixture of discrete distributions, using Dirichlet beliefs over the mixture weights. We study a variation of this concept, in which the documents' mixture weight beliefs are replaced with squashed Gaussian distributions. This allows documents to be associated with elements of a Hilbert space,
admitting kernel topic models (KTM), modelling temporal, spatial, hierarchical, social and other structure between documents. The main challenge is efficient approximate inference on the latent Gaussian. We present an approximate algorithm cast around a Laplace approximation in a transformed basis. The KTM can also be interpreted as a type of Gaussian process latent variable model, or as
a topic model conditional on document features, uncovering links between earlier work in these areas.

We present a method for fully automated selection of treatment beam ensembles for external radiation therapy. We reformulate the beam angle selection problem as a clustering problem of locally ideal beam orientations distributed on the unit sphere. For this purpose we construct an infinite mixture of von Mises-Fisher distributions, which is suited in general for density estimation from data on the D-dimensional sphere. Using a nonparametric Dirichlet process prior, our model infers probability distributions over both the number of clusters and their parameter values. We describe an efficient Markov chain Monte Carlo inference algorithm for posterior inference from experimental data in this model. The performance of the suggested beam angle selection framework is illustrated for one intra-cranial, pancreas, and prostate case each. The infinite von Mises-Fisher mixture model (iMFMM) creates between 18 and 32 clusters, depending on the patient anatomy. This suggests to use the iMFMM directly for beam ensemble selection in robotic radio surgery, or to generate low-dimensional input for both subsequent optimization of trajectories for arc therapy and beam ensemble selection for conventional radiation therapy.

Round-based games are an instance of discrete planning problems. Some of the best contemporary game tree search algorithms use random roll-outs as data. Relying on a good policy, they learn on-policy values by propagating information upwards in the tree, but not between sibling nodes. Here, we present a generative model and a corresponding approximate message passing scheme for inference on the optimal, off-policy value of nodes in smooth AND/OR trees, given random roll-outs. The crucial insight is that the distribution of values in game trees is not completely arbitrary. We define a generative model of the on-policy values using a latent score for each state, representing the value under the random roll-out policy. Inference on the values under the optimal policy separates into an inductive, pre-data step and a deductive, post-data part. Both can be solved approximately with Expectation Propagation, allowing off-policy value inference for any node in the (exponentially big) tree in linear time.

Our goal is to understand the principles of Perception, Action and Learning in autonomous systems that successfully interact with complex environments and to use this understanding to design future systems